Uncertainty Computation at Finite Distance in Nonlinear Mixed Effects Models-a New Method Based on Metropolis-Hastings Algorithm.

Markov Chain Monte Carlo (MCMC) is widely applied to fit complex, practical models in various disciplines, such as engineering, physics, agriculture, etc. Indeed, when the target density $f$ is not available in closed form, a Markov chain is constructed with stationary density $f$ for forming Monte Carlo estimators of means with respect to $f$. Although a Harris ergodic Markov chain will converge to the target distribution as the Monte Carlo sample size tends to infinity, and the population means can be consistently estimated by the corresponding MCMC estimators, in practice, it is important to estimate the standard errors associated with these Monte Carlo estimates due to the availability of only a finite sample. Therefore, the geometric ergodicity of a Markov Chain is highly desired as it is the most standard method for guaranteeing a central limit theorem (CLT) for MCMC estimators and finding its standard errors. Among the different MCMC algorithms, Metropolis-Hastings (MH) algorithms are predominant. This thesis establishes conditions under which MH algorithms with a position-dependent proposal covariance matrix will or will not have the geometric convergence rate. Indeed, we provide these results for MH algorithms where the normal proposal density has a general mean function $c(x)$ and a covariance matrix $G(x)$ depending on the current position $x$ of the Markov chain. For modern variants of the popular Metropolis adjusted Langevin algorithms (MALA), like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for the geometric ergodicity of different variations of the Langevin algorithms, e.g., MMALA, position-dependent MALA (PMALA), and the pre-conditioned MALA (PCMALA). We also provide conditions guaranteeing geometric ergodicity of the pre-conditioned unadjusted Langevin algorithms (PCULA). Generalized linear mixed models (GLMMs), where the response variable is allowed to follow a non-Gaussian distribution, are popular statistical models, although their likelihood functions are not available in closed form. GLMMs can handle the over-dispersion often present in real-world data, and MCMC methods are generally used to fit frequentist and Bayesian GLMMs. Since the likelihood function for GLMMs is not available in closed form, Monte Carlo EM or Monte Carlo maximum likelihood methods are often used for making inferences from GLMMs. Simulation from the conditional (on the data) distribution of the random effects is, in turn, needed for implementing the Monte Carlo EM or Monte Carlo maximum likelihood methods. In this thesis, our general conditions for geometric ergodicity of variants of Langevin algorithms are also verified in the context of conditional simulation from the two most popular GLMMs, namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. These results have important practical implications as users can now confidently fit these models by attaching asymptotically valid standard errors with their Markov chain-based Monte Carlo estimates. Spatial generalized linear mixed models (SGLMMs) are GLMMs where the random effects form a spatial process and are useful for modeling spatially correlated data. We compare the performance of different variants of MALA mentioned above as well as other popular MH algorithms like Random Walk Metropolis in the context of conditional simulation from SGLMMs. Next, we consider MCMC algorithms for full Bayesian analysis of GLMMs. Blocking and reparameterization are techniques used to improve the efficiency and convergence properties of MCMC algorithms. We combine these strategies with pre-conditioning to develop efficient Langevin algorithms for Bayesian GLMMs and SGLMMs. We illustrate our algorithms using extensive real and simulated data analyses.

The aim of this study was to investigate the use of nonlinear mixed effects (NLME) models in a real bioequivalence study and compare it to noncompartmental analysis (NCA) which is proposed by regulatory agencies. NCA requires few hypotheses but a large number of samples per subject. On the other hand, NLME approach is more complex than NCA but it has some advantages such as it requires few samples per subject. A real data application was provided for the study, which was get from Ege University Drug Development and Pharmacokinetics Research Center. For NLME, we used the stochastic approach to expectation-maximization (SAEM) algorithm, whereas linear trapezoidal rule was used for NCA. We estimate pharmacokinetic parameters, area under the curve (AUC0-∞) and maximum concentration (Cmax), and perform a bioequivalence tests using NCA and NLME. According to real data analysis, NLME approach has smaller within subject error, narrower confidence intervals than non-compartmental analysis. However, NLME models have some limitations because of increasing type I error. Therefore, caution is needed for small sample size and data with high variability.

Extending the code in the open-source saemix package to fit joint models of longitudinal and time-to-event data

Confidence intervals for the maximum likelihood estimates (MLEs) are commonly used in statistical inference. To accurately construct such confidence intervals, one typically needs to know the distribution of the MLE. Standard statistical theory says normalized MLE is asymptotically normal with mean zero and variance being a function of the Fisher Information Matrix (FIM) at the unknown parameter. Two common estimates for the variance of MLE are the observed FIM (same as Hessian of negative log-likelihood) and the expected FIM, both of which are evaluated at the MLE given sample data. We show that, under reasonable conditions, the expected FIM tends to outperform the observed FIM under a mean-squared error criterion. This result suggests that, under certain conditions, the expected FIM is a better estimate for the variance of MLE when used in confidence interval calculations.

An individual-tree diameter growth model was developed for Cunninghamia lanceolata in Fujian province, southeast China. Data were obtained from 72 plantation-grown China-fir trees in 24 single-species plots. Ordinary non-linear least squares regression was used to choose the best base model from among 5 theoretical growth equations; selection criteria were the smallest absolute mean residual and root mean square error and the largest adjusted coefficient of determination. To account for autocorrelation in the repeated-measures data, we developed one-level and nested two-level nonlinear mixed-effects (NLME) models, constructed on the selected base model; the NLME models incorporated random effects of the tree and plot. The best random-effects combinations for the NLME models were identified by Akaike's information criterion, Bayesian information criterion and −2 logarithm likelihood. Heteroscedasticity was reduced with two residual variance functions, a power function and an exponential function. The autocorrelation was addressed with three residual autocorrelation structures: a first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and a compound symmetry structure (CS). The one-level (tree) NLME model performed best. Independent validation data were used to test the performance of the models and to demonstrate the advantage of calibrating the NLME models.

In this study we compared the efficiency of frequentist approach with Bayesian approach by carrying out extreme value analysis of Annual Maximum Daily Rainfall (AMDR). For frequentist frequency analysis of AMDR, we used the data of one station i.e. Lahore in Punjab province, Pakistan while for Bayesian analysis we used the data of three other neighboring stations as prior information. During frequentist approach, Generalized Extreme Value (GEV) was found to be a best-fit distribution on the data. In frequentist method, the parameters of GEV distribution were estimated using Maximum Likelihood Estimation (MLE), while in the Bayesian framework the Markov Chain Monte Carlo (MCMC) simulation technique along with Metropolis-Hasting algorithm and Gibbs sampler were implemented. Findings of this study indicate that despite the asymptotic requirements of the MLE, its performance can be enhanced by adopting the Bayesian MCMC paradigm. In order to acquire the posterior densities of GEV parameters, non-informative and informative priors based on the historical data of surrounding weather stations were employed. The result of Bayesian MCMC might be affected by the choice of priors. In addition, the performance of the parameters estimation methods was verified by employing several robustness measures. Robustness measures results proved that the Bayesian MCMC method performed better than MLE in estimating GEV parameters and future return levels. Therefore, the findings of these analyses could be helpful in adopting proper flood protection measures and designing infrastructures of culverts, buildings, bridges and dams in the region.

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

Some bootstrap methods in nonlinear mixed-effect models

Different mixed-effects models were compared to evaluate the population dose-response and relative potency of two albuterol inhalers. Bronchodilator response was measured after ascending doses of each inhaler in 37 asthmatic patients. A linear mixed-effects model was developed based on the approach proposed by Finney for the evaluation of bioassay data. A nonlinear mixed-effects (Emax) model with interindividual and interoccasion variability (IOV) in the different pharmacodynamic parameters was also fit to the data. Both methods produced a similar estimate of relative potency. However, the estimate of relative potency was 22% lower with the nonlinear mixed-effects model if IOV was not taken into account. Monte Carlo simulations based on a similar study design demonstrated that more biased and variable estimates of ED50 and relative potency were obtained when the nonlinear mixed-effects model ignored the presence of IOV in the data. Furthermore, the linear mixed-effects model that did not account for IOV produced confidence intervals for relative potency that were too narrow and thus could lead to erroneous conclusions. These problems were avoided when the estimation model could account for IOV. Results of the simulations were consistent with those of the experimental data. Although the linear or the nonlinear mixed-effects model may be used to evaluate population dose-response and relative potency, there are important differences in the assumptions made by each method.

Locus heterogeneity is a major problem plaguing the mapping of disease genes responsible for complex genetic traits via linkage analysis. A common feature of several available methods to account for heterogeneity is that they involve maximizing a multidimensional likelihood to obtain maximum likelihood estimates. The high dimensionality of the likelihood surface may be due to multiple heterogeneity (mixing) parameters, linkage parameters, and/or regression coefficients corresponding to multiple covariates. Here, we focus on this nontrivial computational aspect of incorporating heterogeneity by considering several likelihood maximization procedures, including the expectation maximization (EM) algorithm and the stochastic expectation maximization (SEM) algorithm. The wide applicability of these procedures is demonstrated first through a general formulation of accounting for heterogeneity, and then by applying them to two specific formulations. Furthermore, our simulation studies as well as an application to the Genetic Analysis Workshop 12 asthma datasets show that, among other observations, SEM performs better than EM. As an aside, we illustrate a limitation of the popular admixture approach for incorporating heterogeneity, proved elsewhere. We also show how to obtain standard errors (SEs) for EM and SEM estimates, using methods available in the literature. These SEs can then be combined with the corresponding estimates to provide confidence intervals of the parameters.

IntroductionAccurate estimation of individual tree aboveground biomass (AGB) is essential for tree species selection, carbon accounting, and precision forestry. Unmanned aerial vehicle (UAV) LiDAR provides rapid access to detailed tree structural information, offering a promising tool for high-frequency biomass assessment.MethodsIn this study, a nonlinear mixed-effects (NLME) model integrating UAV LiDAR and field measurements was developed to quantify the influence of genetic heterogeneity and environmental factors on AGB estimation of Catalpa bungei. Data from 2,941 trees across 79 genotypes were collected in Henan Province, including LiDAR-derived tree height (LH), LiDAR-derived crown diameter (LCD), and AGB. By incorporating genotype as a random effect and planting density as a dummy variable, the NLME model significantly outperformed traditional dummy-variable models.ResultsGenotype effects explained significant AGB variation, achieving high accuracy (R²=0.7916, RMSE = 3.7095) and reducing TRE by 23.29% compared to the basic power function model. Leave-one-genotype-out cross-validation confirmed robustness. Calibration with the four largest trees yielded the best performance (TRE = 13.09%), while a simplified scheme using only two trees per genotype maintained high accuracy (TRE = 13.24%), markedly reducing field effort.DiscussionThese results highlight the superiority of NLME AGB models over linear approaches and demonstrate that accounting for genotype effects is critical for reliable biomass estimation. The proposed framework provides an efficient and cost-effective solution for biomass monitoring, tree breeding, carbon sink assessment, and precision forestry.

PFIM 4.0, an extended R program for design evaluation and optimization in nonlinear mixed-effect models

This work is devoted to the statistical approach to ocean acoustic tomography. Maximum-likelihood (ML) estimate of sound speed field is considered. Deterministic, quasi-deterministic and gaussian signals in the presence of gaussian background noise are examined. A rigorous expression for the functional derivative of the Green function with respect to the sound speed field is applied to find the ML estimate. Fisher information matrices (FIM) giving the Cramer-Rao bounds are obtained for various signal models. The modal, ray and interference tomography methods for global, large scale and small scale inhomogeneities are derived using the statistical approach. FIM of estimate of mode or ray delay are obtained. Some numerical evaluations of potential accuracy of sound-speed field reconstruction in typical propagation conditions are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Methods and software tools for optimal design in nonlinear mixed effect models, based on the Fisher information matrix, have been developed for a decade.1,2 Academic groups regularly proposed new versions.3–5 Present tools do not incorporate adaptive designs for these models although prior information is needed and adaptive designs are increasingly used in drug development.6 We conducted a study among drug companies of the Drug and Disease Model Resources consortium7 to identify current practices and expectations.CPT: Pharmacometrics &amp; Systems Pharmacology (2013) 2, e46; doi:10.1038/psp.2013.19; advance online publication 5 June 2013

In the current study, a novel approach combining quantile regression with nonlinear mixed-effects (QR-NLME) modeling was applied to predict the maximum crown width (cwmax) of three economically important forest species—the European beech (Fagus sylvatica L.), the black pine (Pinus nigra Arn.), and the Calabrian pine (Pinus brutia Ten.) at tree level. A power QR-NLME model was fitted first to a dataset including 1414 European beech trees obtained from 29 randomly distributed sample plots, 770 black pine trees from 25 sample plots, and 1880 Calabrian pine trees from 41 sample plots in Greece, to predict the cwmax at tree level. Additionally, a nonlinear mixed-effects model (NLME) was fitted to the same dataset to predict the average crown width at tree level for all species. In the second stage, the crown competition factor (CCF) was estimated based on the population average response of the cwmax predictions. The proposed approach presented sound results when compared with the outcomes of relevant models from other regions fitted to open-grown tree data, and therefore, it can be well implemented on clustered data structures, in cases of absence of open-grown tree data.